We propose a physical framework for ant navigation of chemical trails. For this, we use controlled experiments in which individuals follow narrow pheromone trails, for which ants display oscillatory motion, as previously reported in the literature. We model this behavior by treating chemotaxis as an effective magnetic interaction between the ants’ velocity and the local chemical gradient. Under suitable approximations, the model yields an analytical expression for the velocity correlations in the direction perpendicular to the trail, predicting an underdamped oscillatory decay. This theoretical prediction is in qualitative agreement with our experimental measurements, indicating that the model captures the essential dynamical features of ant trail following. We fit the model parameters to individual trajectories in order to assess the consistency of the underlying assumptions, finding the same parameter relationship in both theory and experiment. Our results contribute to the characterization of chemotactic navigation in ants and illustrate how physical modeling can provide mechanistic insights into complex biological dynamics.

Understanding the fitness advantages conferred by eusociality remains a central challenge in behavioral ecology. One promising approach is to identify collective strategies that shift efficiency within social groups. Here, we test the hypothesis that reserve workforces in eusocial insect colonies represent an adaptive mechanism that enhances flexibility and foraging efficiency under fluctuating environmental conditions. We examine how such reserve workers modulate the departure and return rates of foragers and how these time-dependent dynamics shape the colony’s overall energetic balance. By integrating an energetic-balance framework with stochastic search simulations inspired by empirical results from Aphaenogaster senilis , we quantify the energetic requirements for colony viability, incorporating energy intake, search costs, and basal metabolic demands. Our results show that as colonies grow, maintaining a positive energy balance requires a disproportionately larger relative workforce. By modulating departure and return rates over time, colonies control the synchrony of their collective search and efficiently activate or suppress their reserve workforce to scale foraging effort as needed. These findings suggest that the “lazy” or weakly engaged workers commonly observed in large colonies function as an essential reserve that stabilizes colony energetics and enhances responsiveness. Together, our results provide a functional explanation for sublinear metabolic scaling in eusocial groups and highlight workforce modulation as a key factor underlying their energetic stability and evolutionary success.

Dispersal is often used by living beings to gather information from conspecifics, integrating it with personal experience to guide decision-making. This mechanism has only recently been studied experimentally, facilitated by advancements in tracking animal groups over extended periods. Such studies enable the analysis of the adaptive dynamics underlying sequential decisions and collective choices. Here, we present a theoretical framework based on the Voter Model to investigate these processes. The model, originally designed to study opinion or behavioral consensus within groups through imitation, is adapted to include the prospection of others' decisions as a mechanism for updating personal criteria. We demonstrate that several properties of our model (such as average consensus times and polarization dynamic) can be analytically mapped onto those of the classical Voter Model under simplifying assumptions. Finally, we discuss the potential of this framework for studying more complex scenarios.

We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which it does not react to the target. We demonstrate that Poissonian resetting leads to the existence of a non-equilibrium steady state. We calculate the distribution of the first arrival time to a target along with its mean and show the existence of an optimal reset rate. In particular, we prove that the initial condition of the walker, i.e., either starting diffusely or relocating, can significantly affect the long-time properties of the search process. Moreover, we demonstrate the presence of distinct parameter regimes for the global optimization of the mean first arrival time when ballistic and diffusive movements are in direct competition.

We use complex systems science to explore the emergent behavioral patterns that typify eusocial species, using collective ant foraging as a paradigmatic example. Our particular aim is to provide a methodology to quantify how the collective orchestration of foraging provides functional advantages to ant colonies. For this, we combine (i) a purpose-built experimental arena replicating ant foraging across realistic spatial and temporal scales, and (ii) a set of analytical tools, grounded in information theory and spin-glass approaches, to explore the resulting data. This combined approach yields computational replicas of the colonies; these are high-dimensional models that store the experimental foraging patterns through a training process, and are then able to generate statistically similar patterns, in an analogous way to machine learning tools. These in silico models are then used to explore the colony performance under different resource availability scenarios. Our findings highlight how replicas of the colonies trained under constant and predictable experimental food conditions exhibit heightened foraging efficiencies, manifested in reduced times for food discovery and gathering, and accelerated transmission of information under similar conditions. However, these same replicas demonstrate a lack of resilience when faced with new foraging conditions. Conversely, replicas of colonies trained under fluctuating and uncertain food conditions reveal lower efficiencies at specific environments but increased resilience to shifts in food location.

We study the effect of a resetting point randomly distributed around the origin on the mean first passage time of a Brownian searcher moving in one dimension. We compare the search efficiency with that corresponding to reset to the origin and find that the mean first passage time of the latter can be larger or smaller than the distributed case, depending on whether the resetting points are symmetrically or asymmetrically distributed. In particular, we prove the existence of an optimal reset rate that minimizes the mean first-passage time for distributed resetting to a finite interval if the target is located outside this interval. When the target position belongs to the resetting interval or it is infinite then no optimal reset rate exists, but there is an optimal resetting interval width or resetting characteristic scale which minimizes the mean first-passage time. We also show that the first-passage density averaged over the resetting points depends on its first moment only. As a consequence, there is an equivalent point such that the first-passage problem with resetting to that point is statistically equivalent to the case of distributed resetting. We end our study by analyzing the fluctuations of the first-passage times for these cases. All our analytical results are verified through numerical simulations.

We explore the case of a group of random walkers looking for a target randomly located in space, such that the number of walkers is not constant but new ones can join the search, or those that are active can abandon it, with constant rates r_{b} and r_{d}, respectively. Exact analytical solutions are provided both for the fastest-first-passage time and for the collective time cost required to reach the target, for the exemplifying case of Brownian walkers with r_{d}=0. We prove that even for such a simple situation there exists an optimal rate r_{b} at which walkers should join the search to minimize the collective search costs. We discuss how these results open a new line to understand the optimal regulation in searches conducted through multiparticle random walks, e.g., in chemical or biological processes.

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward B and forward F times being the times since the last and until the next reset, respectively, given that the current state of the system X(t) is known. These magnitudes are introduced with the paradigmatic case of diffusion under resetting, for which the backward and forward times are conditioned to the position of the walker. We find analytical expressions for the conditioned backward and forward time probability density functions (PDFs), and we compare them with numerical simulations. The general expressions allow us to study particular scenarios. For instance, for power-law reset time PDFs such that φ(t)∼t^{-1-α}, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of α due to the composition between the long-time scaling of diffusion P(x,t)∼1/sqrt[t] and the reset time PDF.

While approaches based on physical grounds (such as the drift-diffusion model-DDM) have been exhaustively used in psychology and neuroscience to describe perceptual decision making in humans, similar approaches to complex situations, such as sequential (tree-like) decisions, are still scarce. For such scenarios that involve a reflective prospection of future options, we offer a plausible mechanism based on the idea that subjects can carry out an internal computation of the uncertainty about the different options available, which is computed through the corresponding Shannon entropy. When the amount of information gathered through sensory evidence is enough to reach a given threshold in the entropy, this will trigger the decision. Experimental evidence in favor of this entropy-based mechanism was provided by exploring human performance during navigation through a maze on a computer screen monitored with the help of eye trackers. In particular, our analysis allows us to prove that (i) prospection is effectively used by humans during such navigation tasks, and an indirect quantification of the level of prospection used is attainable; in addition, (ii) the distribution of decision times during the task exhibits power-law tails, a feature that our entropy-based mechanism is able to explain, unlike traditional (DDM-like) frameworks.

We study a non-Markovian and nonstationary model of animal mobility incorporating both exploration and memory in the form of preferential returns. Exact results for the probability of visiting a given number of sites are derived and a practical WKB approximation to treat the nonstationary problem is developed. A mean-field version of this model, first suggested by Song et al., [Modelling the scaling properties of human mobility, Nat. Phys. 6, 818 (2010)NPAHAX1745-247310.1038/nphys1760] was shown to well describe human movement data. We show that our generalized model adequately describes empirical movement data of Egyptian fruit bats (Rousettus aegyptiacus) when accounting for interindividual variation in the population. We also study the probability of visiting any site a given number of times and derive a mean-field equation. Our analysis yields a remarkable phase transition occurring at preferential returns which scale linearly with past visits. Following empirical evidence, we suggest that this phase transition reflects a trade-off between extensive and intensive foraging modes.

We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results, we find that when the waiting time probability has first and second finite moments, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset point to the boundary, or (iii) always beneficial. Instead, when the waiting time probability has the first or the two first moments diverging we find that resetting is always beneficial. Finally, we have also found that the optimal strategy to exit the domain depends on the reset rate. For low reset rates, walkers with exponential waiting times are found to be optimal, while for high reset rate, anomalous waiting times optimize the search process.

While perceptual decision making in humans is often considered to be governed by evidence accumulator models (like drift-diffusion), mechanisms driving harder situations where prospection of future scenarios is necessary remain largely unknown. Here, experimental and computational evidence is given in favour of a mechanism in which prospection of possible future payoffs associated to each available choice could be used, through the internal estimation of the corresponding Shannon's entropy $S$. So, the decision would be triggered as soon as $S$ reaches a threshold which ensures that a choice is reliable enough. We illustrate this idea using a task in which subjects have to navigate sequentially through a maze on the computer screen while avoiding trajectory overlaps, forcing them to use memory and prospection skills that we indirectly capture through eye-tracking. Comparison of the experimental data to that from virtual (ideal) subjects allows us to verify that the performances observed, as well as the distribution of decision-making times, of humans are compatible with the aforementioned mechanism.

Information on the relevant global scales of the search space, even if partial, should conceivably enhance the performance of random searches. Here we show numerically and analytically that the paradigmatic uninformed optimal Lévy searches can be outperformed by informed multiple-scale random searches in one (1D) and two (2D) dimensions, even when the knowledge about the relevant landscape scales is incomplete. We show in the low-density nondestructive regime that the optimal efficiency of biexponential searches that incorporate all key scales of the 1D landscape of size L decays asymptotically as η_{opt}∼1/sqrt[L], overcoming the result η_{opt}∼1/(sqrt[L]lnL) of optimal Lévy searches. We further characterize the level of limited information the searcher can have on these scales. We obtain the phase diagram of bi- and triexponential searches in 1D and 2D. Remarkably, even for a certain degree of lack of information, partially informed searches can still outperform optimal Lévy searches. We discuss our results in connection with the foraging problem.

Self-avoidance is a common mechanism to improve the efficiency of a random walker for covering a spatial domain. However, how this efficiency decreases when self-avoidance is impaired or limited by other processes has remained largely unexplored. Here we provide a numerical study in regular lattices for the case when the self-avoiding signal left by a walker both (i) saturates after successive revisits to a site, and (ii) evaporates, or disappears, after some characteristic time. We surprisingly reveal that the mean cover time becomes minimum for intermediate values of the evaporation time, leading to the existence of a nontrivial optimum management of the self-avoiding signal. We show that this is a consequence of a complex dynamics arising from the interplay between signal evaporation and signal saturation, in which evaporation has the capacity of creating some sort of mirages (sites or regions that the walker see as unvisited, though in fact they are not) that enhance the searcher mobility, so contributing to a more efficient exploration of the lattice that counteracts the effects of signal saturation. Remarkably, we argue both through scaling arguments and from numerical results, that this mirage effect will become more and more significant as long as the domain size increases.

Excited random walks represent a convenient model to study food intake in a media which is progressively depleted by the walker. Trajectories in the model alternate between (i) feeding and (ii) escape (when food is missed and so it must be found again) periods, each governed by different movement rules. Here, we explore the case where the escape dynamics is adaptive, so at short times an area-restricted search is carried out, and a switch to extensive or ballistic motion occurs later if necessary. We derive for this case explicit analytical expressions of the mean escape time and the asymptotic growth of the depleted region in one dimension. These, together with numerical results in two dimensions, provide surprising evidence that ballistic searches are detrimental in such scenarios, a result which could explain why ballistic movement is barely observed in animal searches at microscopic and millimetric scales, therefore providing significant implications for biological foraging.

Information gathering and processing can be particularly useful to enhance the efficiency of living organisms when searching or foraging under uncertain scenarios. Assuming that such situations can be conveniently modeled through random walk models, a case that has been widely explored in the physical literature is the generation of self-avoiding trajectories as a mechanism to increase the area of the region explored and so the search efficiency. Additionally, other mechanisms as the ability to prospect the possible outcomes from choosing between different paths/trajectories could potentially enhance such efficiency even further. Here we use an extension of the classical true self-avoiding random walk model to explore that idea. We evaluate numerically how the coverage time of such random walk model gets modified when a basic prospection mechanism, which allows the walker to anticipate the outcome of possible future paths, is introduced. We analyze the level of prospection that is required to optimize the mean coverage time in a regular domain and report two relevant findings. First, we observe that increasing more and more the information used for prospection is just optimal provided the organism can process it adequately (which occurs when the self-avoidance mechanism is reliable enough), but it can be even detrimental when self-avoidance (i.e. memory) is poor or impaired. Second, we surprisingly find that prospection carried out at different time scales can be as optimal as one based on a fixed large prospection time scale. We claim that such results may capture some basic traits from the real strategies used by higher organisms when searching under uncertainty.

We investigate the effects of Markovian resetting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power-law probability density functions. We prove the existence of a nonequilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power-law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target, and the characteristic transport exponents.

There is a widespread belief in ecology that the capacity of animals to orchestrate systematic and planned paths should represent a significant benefit for efficient search and exploration. Within this view, stochasticity observed in real animal trajectories is mostly understood as undesirable noise caused by internal or external effects. Far less is known, however, about the case when cognitive errors and limitations inherent to living systems are explicitly put into play. Here we compare within this context the search efficiency of (i) walkers driven by Bayesian rules generating deterministic paths, (ii) standard random walkers, and (iii) human trajectories obtained from search experiments in a soccer field and on the computer screen. Our results clearly challenge the view that deterministic paths are generally better for exploration than random strategies, as the latter are more resilient to cognitive errors. Instead, we provide numerical and experimental evidence that stochasticity would provide living organisms with a sufficient and cognitively simple exploration solution to the problem of uncertainty.

Emergence of collective, as well as superorganism-like, behaviour in biological populations requires the existence of rules of communication, either direct or indirect, between organisms. Because reaching an understanding of such rules at the individual level can be often difficult, approaches carried out at higher, or effective, levels of description can represent a useful alternative. In the present work, we show how a spin-glass approach characteristic of statistical physics can be used as a tool to characterize the properties of the spatial occupancy patterns of a biological population. We exploit the presence of pairwise interactions in spin-glass models for detecting correlations between occupancies at different sites in the media. Such correlations, we claim, represent a proxy to the existence of planned and/or social strategies in the spatial organization of the population. Our spin-glass approach does not only identify those correlations but produces a statistical replica of the system (at the level of occupancy patterns) that can be subsequently used for testing alternative conditions/hypothesis. Here, this methodology is presented and illustrated for a particular case of study: we analyse occupancy patterns of Aphaenogaster senilis ants during foraging through a simplified environment consisting of a discrete (tree-like) artificial lattice. Our spin-glass approach consistently reproduces the experimental occupancy patterns across time, and besides, an intuitive biological interpretation of the parameters is attainable. Likewise, we prove that pairwise correlations are important for reproducing these dynamics by showing how a null model, where such correlations are neglected, would perform much worse; this provides a solid evidence to the existence of superorganism-like strategies in the colony.

In this work, we consider stochastic movement with random resets to the origin followed by a random residence time before the motion starts again. First, we study the transport properties of the walker, i.e. we derive an expression for the mean square displacement of the overall process and study its dependence on the statistical properties of the resets and the residence times probability density functions (PDFs) and the type of movement. From this general formula, we see that the inclusion of the residence after the resets is able to induce super-diffusive to sub-diffusive (or diffusive) regimes and it can also make a sub-diffusive walker reach a constant mean square displacement or even collapse. Second, we study how the reset-and-residence mechanism affects the survival probability of different search processes to a given position, showing that the long time behavior of the reset and residence time PDFs determine the existence of the mean first arrival time.